Beat_(acoustics) - Wikiparency

# Beat (acoustics) > *For the beats heard between loud tones that are not present in the air, see [[Combination_tone]].* A **beat** is the slow, regular swelling and fading of loudness heard when two [[Sound|sounds]] of slightly different [[Frequency|frequency]] are played together. The two pressure waves drift in and out of step: when their crests coincide they add, and half a beat later a crest of one meets a trough of the other and they cancel. The listener hears a single tone at the average of the two frequencies whose loudness rises and falls |f₁ − f₂| times a second, the **beat frequency**. Beats are the most familiar audible case of [[Wave_interference|wave interference]] and of the [[Superposition_principle|superposition principle]]. Beats are a working tool as well as a curiosity. Piano tuners and orchestral players listen for them and adjust a string or a slide until they slow to nothing, and the equal-tempered scale is defined in practice by the beat rates of its slightly impure intervals. When the two tones are fed one to each ear through headphones, a fainter *binaural* beat is heard, which is produced inside the brain rather than in the air. The Wikitube microsim for this article plays two tones side by side: the reader sets the first tone's frequency, the gap between the two and their amplitude ratio, and watches the zoomed waveforms slide past one another, the long trace swell and fade under its envelope, and the two spectral lines that stand behind it. ## Mathematics and physics of beat tones Two tones of equal amplitude, p₁ = sin(2πf₁t) and p₂ = sin(2πf₂t), add at a point in space to a single product by the sum-to-product identity: `p = sin(2πf₁t) + sin(2πf₂t) = 2 cos(2π·(f₁ − f₂)/2·t) · sin(2π·(f₁ + f₂)/2·t)` The second factor is a fast [[Sine_wave|sine wave]] at the mean frequency (f₁ + f₂)/2, the *carrier*, which the ear hears as the pitch. The first factor is a slow cosine at half the difference frequency, which acts as an envelope. The loudness depends on the size of the envelope, not its sign, so the loudness peaks twice per cycle of the cosine, and the beat frequency is the full difference, f_beat = |f₁ − f₂|. OpenStax gives the same result as its Eq. 17.17 and notes that a negative beat frequency has no meaning; its worked example of two tuning forks at 256 Hz and 512 Hz gives 256 Hz, far too fast to hear as a beat.[^ost-176] With the microsim at its defaults, f₁ = 440 Hz and f₂ = 446 Hz, the carrier sits at 443 Hz and the loudness swells six times a second: one full beat every 1/6 s, or about 167 ms. The top panel of the sim shows 10 ms of the two tones strobed on tone 1, so tone 2 appears to crawl past it at the difference frequency; the middle panel shows the first half-second of the sum, three complete beats under an orange envelope that pinches to zero at each cancellation; and the bottom panel is the [[Frequency_domain|frequency-domain]] view, two lines 6 Hz apart. The sum contains no energy at the beat frequency itself. The beat is a slow change of level of the 443 Hz carrier, which is why a spectrum analyzer shows only the two original lines.[^thinkdsp-1] When the amplitudes are unequal, p = sin(2πf₁t) + a·sin(2πf₂t) with a < 1, the cancellation is never complete. The envelope swings between 1 + a and 1 − a, and the depth of the beat in decibels is 20 log₁₀[(1 + a)/(1 − a)]. For a = 0.5 the level changes by about 9.5 dB between swell and trough; for a = 0.8 by about 19 dB; only for a = 1 does the trough fall to silence. Beats are therefore most obvious between two sources of similar loudness, such as two strings of the same piano note or two voices singing in unison. The same algebra holds for any pair of waves that add linearly, whether on a string, in air or on a radio carrier, which ties beats to the [[Group_velocity|group velocity]] of a wave packet: a packet is a long sum of close frequencies, and its envelope moves as the beat pattern of those frequencies does.[^ost-176] *Try: set f2 − f1 to 6 Hz and count the swells in the long trace; drag it down to zero and watch the zoom stop sliding; lower the amplitude ratio a and the pinches become shallow; raise f2 − f1 past 15 Hz and the caption switches from beats to roughness.* ## Monaural beats Monaural beats are the ordinary kind: both tones reach the same ear, or both ears, through the air, and they add physically before the ear does any processing. The [[Hearing|ear]] follows the swelling as a pulsation in loudness while the difference is a few hertz; the microsim switches its caption at 15 Hz as an illustrative threshold. As the difference grows toward a few tens of hertz the pulsations blur into a sensation of *roughness*, a harsh or buzzing quality, and when the difference grows larger than the ear's critical band the listener stops hearing one fluctuating tone and hears two separate pitches. Plomp and Levelt measured this sequence with pairs of pure tones in 1965: the rated dissonance of two tones rises steeply from unison, peaks at a separation of about a quarter of a critical bandwidth, and falls away as the two tones separate into distinct pitches.[^plomp] The grey curve in the microsim's spectrum panel is a fit to that data in the form Sethares published in 1993, drawn as an illustration rather than as a measurement.[^sethares] The transition from beats to roughness is the physical root of [[Consonance_and_dissonance|consonance and dissonance]] between complex tones. A musical note is not one sine wave but a [[Fundamental_frequency|fundamental]] with a series of [[Harmonic|harmonics]], and two notes a fifth apart share harmonics: the third harmonic of the lower note coincides with the second harmonic of the upper. If the interval is tuned exactly to the ratio 3:2, the coinciding harmonics lock together and nothing beats. If it is slightly mistuned, those harmonics beat slowly, and the beat rate tells the musician how far off the interval is. Roughness between the non-coinciding harmonics of two notes is what makes some intervals sound smoother than others, and the pattern of coinciding [[Overtone|overtones]] is why the same interval can sound different on instruments of different [[Timbre|timbre]]. A related effect, the [[Combination_tone|combination tone]], arises when two loud tones are heard together and the ear itself, being slightly nonlinear, adds new tones such as f₂ − f₁ that are not in the air. Beats need no such nonlinearity. The beat is present in the pressure waveform that reaches the eardrum and would be recorded by any [[Microphone|microphone]]. ## Binaural beats When the two tones are delivered separately, one to each ear through headphones, they never add in the air, yet many listeners hear a faint, slow fluctuation that seems to move inside the head. This is a binaural beat. Gerald Oster, whose 1973 *Scientific American* article revived interest in the effect, dated its discovery to 1839 and the German physicist H. W. Dove. Oster reported that binaural beats are clearest when the carrier is near 440 Hz, fade at higher frequencies and vanish above about 1,000 Hz, and form only when the two tones differ by less than about 30 Hz.[^oster] He placed the interaction at the superior olivary nucleus of the brainstem, the first stage of the [[Auditory_system|auditory pathway]] to receive signals from both ears.[^oster] The frequency limit fits that explanation. A binaural beat depends on the brain comparing the timing of the nerve impulses from the two ears, and auditory nerve fibers follow the phase of a tone well only at low frequencies; the same phase comparison lets a listener judge the direction of a low-pitched sound from the tiny interaural time difference, the subject of [[Sound_localization|sound localization]]. A binaural beat is in effect a phase difference between the ears that keeps turning over, heard as a sound that wanders from side to side. Binaural beats are widely marketed as a way to entrain brain rhythms and alter mood, attention or sleep. The evidence is mixed. A 2023 systematic review of 14 studies that measured brain oscillations during binaural-beat stimulation found five whose results fit the entrainment hypothesis, eight whose results contradicted it and one with mixed results, and concluded that the outcomes were inconsistent and that study methods needed standardizing.[^ingendoh] An earlier meta-analysis of behavioral outcomes reported small-to-moderate effects on anxiety, memory, attention and pain, with wide variation between studies.[^garcia] The microsim plays monaural beats only; the physics of a binaural beat happens in the listener's head, not in the waveform on screen. ## Uses The oldest use is tuning. OpenStax describes the procedure a piano tuner follows: strike a [[Tuning_fork|tuning fork]], play the matching piano note, and adjust the string until the beats slow down and stop.[^ost-176] The reference pitch for most Western music, A = 440 Hz, was set as an international standard in ISO 16:1975.[^iso16] Unison strings are tuned the same way, by removing beats entirely. Other intervals are tuned by counting beats rather than removing them, because in equal temperament, the standard [[Musical_tuning|tuning]] of Western keyboards, every interval except the octave is slightly impure: the octave is divided into twelve equal semitones of ratio 2^(1/12), so the fifth and the major third fall a little off the simple ratios 3:2 and 5:4.[^muth-623] A worked example shows the size of the effect. Starting from A₃ = 220 Hz, the equal-tempered E₄ is 220 × 2^(7/12) ≈ 329.63 Hz, just short of the pure fifth at 330 Hz. The third harmonic of A₃ is 660 Hz and the second harmonic of E₄ is 659.26 Hz, so the two coinciding harmonics beat at about 0.74 Hz, a slow wave roughly every 1.3 s. The equal-tempered major third is much worse: C♯₄ = 220 × 2^(4/12) ≈ 277.18 Hz against a pure 275 Hz, and the fifth harmonic of A₃ (1,100 Hz) beats with the fourth harmonic of C♯₄ (1,108.73 Hz) at about 8.7 Hz, a fast flutter. A tuner who sets the temperament by ear counts these rates against a watch. On a real piano the harmonics of a stiff string are slightly sharp of the exact integer multiples, which is one reason practical tuning departs from the textbook figures; that is the subject of [[Piano_acoustics|piano acoustics]] and [[String_vibration|string vibration]]. Beats are used wherever a small frequency difference has to be measured with no instrument better than a comparison. Two [[Oscillation|oscillators]] can be set to the same frequency by adjusting one until the beats vanish, the "zero-beat" method of radio and [[Electronics|electronics]] work. Radio receivers mix an incoming signal with a local oscillator and keep the difference frequency, the same sum-to-product identity at radio frequencies; this is the heterodyne principle of [[Radio-frequency_engineering|radio-frequency engineering]]. The theremin, an electronic instrument of the 1920s, makes its audible tone as the difference between two radio-frequency oscillators, one of which the player detunes by moving a hand near an antenna.[^glinsky] Continuous-wave [[Radar|radar]] and Doppler ultrasound measure speed the same way: the echo, shifted by the [[Doppler_effect|Doppler effect]], is mixed with the transmitted signal and the beat between them is the Doppler frequency. ## Sample The Wikipedia pair carries an audio sample of two close tones. Wikitube's sample is the microsim itself, which computes the waveform exactly and can be read as a recipe. Two sine waves of equal amplitude at 440 Hz and 446 Hz, summed and played for two seconds, give twelve beats: a pure tone near A₄ that swells and fades six times a second. In *Think DSP*'s notation the sample is two `SinSignal` objects added with the `+` operator into a `SumSignal`, evaluated with `make_wave` and written to a file; the book's first chapter builds exactly such a mix of two sinusoids and plots its spectrum.[^thinkdsp-1] Lengthening the sample does not change what is heard. Narrowing the gap to 1 Hz gives one slow swell a second; widening it to 30 Hz gives the rough, buzzing sound that the sim's caption describes; and widening it past a few tens of hertz gives two tones. The sim runs its playhead slower than real time by a factor set with the slow-motion control, so the eye can follow the phase slide that the ear hears as a beat. The time axis and the frequencies in the readout are the true ones; only the moving cursor is slowed, and the header says so. ## See also - [[Combination_tone]] - [[Missing_fundamental]] - [[Wave_interference]] - [[Musical_acoustics]] · [[Noise]] · [[Hearing]] · [[String_vibration]] — the neighboring sections of the Acoustics spine - [[Consonance_and_dissonance]] - [[PORTAL_Acoustics]] ## References [^ost-176]: Ling, Samuel J.; Sanny, Jeff; Moebs, William (2016). *University Physics Volume 1*. OpenStax. §17.6 "Beats," pp. 838–840, Eq. 17.17 and Example 17.7; the wave-packet remark closes the section. https://openstax.org/details/books/university-physics-volume-1 — on the [[PORTAL_Acoustics]] book shelf (077). [^thinkdsp-1]: Downey, Allen B. (2012). *Think DSP: Digital Signal Processing in Python*. Green Tea Press. Ch. 1 "Sounds and signals," §1.2 "Spectral decomposition" and §1.3 "Signals" (the `SumSignal` of two sinusoids, Figure 1.4). https://greenteapress.com/wp/think-dsp/ — on the [[PORTAL_Acoustics]] book shelf (058). [^plomp]: Plomp, R.; Levelt, W. J. M. (1965). "Tonal consonance and critical bandwidth." *Journal of the Acoustical Society of America* 38 (4): 548–560. https://doi.org/10.1121/1.1909741 [^sethares]: Sethares, William A. (1993). "Local consonance and the relationship between timbre and scale." *Journal of the Acoustical Society of America* 94 (3): 1218–1228. https://doi.org/10.1121/1.408175 [^oster]: Oster, Gerald (October 1973). "Auditory beats in the brain." *Scientific American* 229 (4): 94–102. https://doi.org/10.1038/scientificamerican1073-94 — source for the 1839 discovery by H. W. Dove, the 440 Hz and 1,000 Hz carrier figures, the ~30 Hz limit and the superior olivary nucleus. [^ingendoh]: Ingendoh, Ruth Maria; Posny, Ella S.; Heine, Angela (May 19, 2023). "Binaural beats to entrain the brain? A systematic review of the effects of binaural beat stimulation on brain oscillatory activity, and the implications for psychological research and intervention." *PLOS ONE* 18 (5): e0286023. https://doi.org/10.1371/journal.pone.0286023 [^garcia]: Garcia-Argibay, Miguel; Santed, Miguel A.; Reales, José M. (2019). "Efficacy of binaural auditory beats in cognition, anxiety, and pain perception: a meta-analysis." *Psychological Research* 83: 357–372. https://doi.org/10.1007/s00426-018-1066-8 [^iso16]: International Organization for Standardization (1975). *ISO 16:1975 Acoustics — Standard tuning frequency (Standard musical pitch)*. Geneva: ISO. (No link given: the catalog URL was not checked in this run.) [^muth-623]: Schmidt-Jones, Catherine (2013). *Understanding Basic Music Theory*. OpenStax CNX. §6.2.3.2 "Equal Temperament" and §6.2.4 "A Comparison of Equal Temperament with the Harmonic Series," pp. 229–231. — on the [[PORTAL_Acoustics]] book shelf (092). [^glinsky]: Glinsky, Albert (2000). *Theremin: Ether Music and Espionage*. University of Illinois Press. (Book-length history of Leon Theremin and his instrument; cited for the heterodyne principle of the instrument and its 1920s origin; page not checked.)  **Microsim — three.js (Wikitube framework):** *Beat (acoustics)* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/acoustics/Beat_(acoustics).html" data-title="Beat (acoustics)"></div> *Built from `MICROSIM_GUIDE/specs/acoustics/sims/Beat_(acoustics).json`; part of the [[PORTAL_Acoustics|Acoustics portal]] spine (section sims and See-also variants).*  ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Beat_(acoustics)) : [Wikitube](https://en.wikitube.io/wiki/Beat_(acoustics)) - skeleton pinned to revision 1370407074 (2026-09-11).